eprints.soton.ac.uk · 2019. 3. 13. · universityofsouthampton abstract...

UNIVERSITY OF SOUTHAMPTON

FACULTY OF PHYSICAL SCIENCES AND ENGINEERING

Physics

Exploratory Lattice QCD Studies of Rare Kaon Decays

by

Andrew J. Lawson

Thesis for the degree of Doctor of Philosophy

October 2017

mailto:[email protected]

UNIVERSITY OF SOUTHAMPTON

ABSTRACT

FACULTY OF PHYSICAL SCIENCES AND ENGINEERINGPhysics

Doctor of Philosophy

EXPLORATORY LATTICE QCD STUDIES OF RARE KAON DECAYS

by Andrew J. Lawson

The rare kaon decays K → π`+`− and K → πνν̄ proceed via flavour changing neu-tral currents, and are thus heavily suppressed in the Standard Model. This naturalsuppression makes these decays sensitive to the effects of potential new physics. Thesedecays first arise as second-order electroweak processes, hence we are required to eval-uate four-point correlation functions involving two effective operators. The evaluationof such four-point correlation functions presents two key difficulties: the appearance ofunphysical terms in Euclidean-space correlators that grow exponentially as the opera-tors are separated, and the presence of ultra-violet divergences as the operators approacheach other. I present the results of the first exploratory studies of the calculation of thelong-distance contributions to these decays using lattice QCD.

The decays K → π`+`− are completely long-distance dominated; this lattice calcula-tion is thus the first step in providing ab-initio estimates for the amplitudes of thesedecays. Our simulations are performed using the 243×64 domain wall fermion ensembleof the RBC-UKQCD collaboration, with a pion mass of 430(2) MeV, a kaon mass of625(2) MeV, and a valence charm mass of 543(13) MeV. In particular we determine theform factor, V (z), of the K+ (k) → π+ (p) `+`− decay from the lattice at small valuesof z = q2/M2K (where q = k − p), obtaining V (z) = 1.37(36), 0.68(39), 0.96(64) for thethree values of z = −0.5594(12), −1.0530(34), −1.4653(82) respectively.

The decays K+ → π+νν̄ are short-distance dominated, although the long-distance con-tributions represent significant sources of uncertainty. The lattice calculation of the decayamplitudes is made particularly difficult by the presence of ultra-violet divergences in thefour-point correlators. I present the calculation of the renormalised decay amplitudes,using the 163×32 domain wall fermion ensemble of the RBC-UKQCD collaboration, witha pion mass of 421(1)(7) MeV, a kaon mass of 563(1)(9) MeV, and a valence charm massof 863(24) MeV. In particular we find the difference between the perturbative and latticeestimates of the charm contribution to these decays to be ∆Pc = 0.0040(13)(32)(−45).


Contents

Declaration of Authorship xvii

Acknowledgements xix

1 Introduction 1

2 Standard Model 72.1 Particle Content and Interactions . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.1.1 Discrete Symmetries . . . . . . . . . . . . . . . . . . . . . 102.1.1.2 Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . . 122.1.1.3 Conserved Currents . . . . . . . . . . . . . . . . . . . . . 13

2.1.2 Electroweak Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.2.1 Electroweak Symmetry Breaking . . . . . . . . . . . . . . 162.1.2.2 CKM Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Fermi Effective Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Operator Product Expansion . . . . . . . . . . . . . . . . . . . . . 192.2.2 Calculation of Wilson Coefficients . . . . . . . . . . . . . . . . . . . 20

2.2.2.1 Operator Renormalisation and Mixing . . . . . . . . . . . 212.2.2.2 Renormalisation Group Evolution . . . . . . . . . . . . . 222.2.2.3 Flavour Boundaries . . . . . . . . . . . . . . . . . . . . . 23

2.3 Chiral Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Phenomenology of Rare Kaon Decays 273.1 Effective Hamiltonians for Rare Kaon Decays . . . . . . . . . . . . . . . . 28

3.1.1 ∆S = 1 Weak Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 283.1.1.1 Wilson Coefficients . . . . . . . . . . . . . . . . . . . . . . 29

3.1.2 Hamiltonian for K → π`+`− . . . . . . . . . . . . . . . . . . . . . 313.1.3 Hamiltonian for K → πνν̄ . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Theoretical Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.1 KS ,K+ → π`+`− . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.2 KL → π0`+`− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.3 K+ → π+νν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.4 KL → π0νν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Lattice QCD 434.1 Lattice Formulation of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.1 Discretisation of Spacetime . . . . . . . . . . . . . . . . . . . . . . 44

v

vi CONTENTS

4.1.2 Gauge Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.1.3 Naive Fermion Discretisation . . . . . . . . . . . . . . . . . . . . . 464.1.4 Wilson Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.5 Overlap Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.6 Domain Wall Fermions . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.7 Conserved Currents . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Numerical Simulation of Lattice Gauge Theory . . . . . . . . . . . . . . . 524.2.1 Pseudofermion Determinant . . . . . . . . . . . . . . . . . . . . . . 524.2.2 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.1 Construction of Correlators . . . . . . . . . . . . . . . . . . . . . . 544.3.2 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3.2.1 Source Smearing . . . . . . . . . . . . . . . . . . . . . . . 564.3.2.2 Sequential Sources . . . . . . . . . . . . . . . . . . . . . . 574.3.2.3 Random Volume Sources . . . . . . . . . . . . . . . . . . 59

4.3.3 Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3.4 Twisted Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 61

4.4 Extraction of Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4.1 Analysis of Correlation Functions . . . . . . . . . . . . . . . . . . . 61

4.4.1.1 Effective Mass Plots . . . . . . . . . . . . . . . . . . . . . 634.4.1.2 3pt Correlators . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.2 Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4.3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4.3.1 Jackknife Resampling . . . . . . . . . . . . . . . . . . . . 664.4.3.2 Bootstrap Resampling . . . . . . . . . . . . . . . . . . . . 67

5 Rare Kaon Decays on the Lattice 695.1 Operators and Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1.1 Z and γ Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.1.2 W-W diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.1.3 Local Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Correlator Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.1 Continuum Euclidean correlators . . . . . . . . . . . . . . . . . . . 755.2.2 Lattice implementation . . . . . . . . . . . . . . . . . . . . . . . . 775.2.3 Removal of Exponentially Growing Intermediate States . . . . . . . 78

5.2.3.1 Single Pion Intermediate State . . . . . . . . . . . . . . . 795.2.3.2 Two and Three Pion Intermediate States . . . . . . . . . 795.2.3.3 Leptonic and Semileptonic Intermediate States . . . . . . 81

5.2.4 Finite Volume Corrections . . . . . . . . . . . . . . . . . . . . . . . 845.3 Renormalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3.1 Non-Perturbative Renormalisation . . . . . . . . . . . . . . . . . . 855.3.2 Local Operator Renormalisation . . . . . . . . . . . . . . . . . . . 88

5.3.2.1 Renormalisation of HW . . . . . . . . . . . . . . . . . . . 895.3.3 Bilocal Operator Renormalisation . . . . . . . . . . . . . . . . . . . 90

5.3.3.1 RI-SMOM Renormalisation . . . . . . . . . . . . . . . . . 915.3.3.2 Matching to MS Scheme . . . . . . . . . . . . . . . . . . . 93

CONTENTS vii

6 Results of K → π`+`− Simulations 956.1 Details of the Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.2 Setup of the calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.2.1 Details of the Implementation . . . . . . . . . . . . . . . . . . . . . 976.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.3.1 2pt and 3pt Correlators . . . . . . . . . . . . . . . . . . . . . . . . 1006.3.2 4pt correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.3.3 Removal of single-pion exponential: Method 1 . . . . . . . . . . . . 1056.3.4 Removal of single-pion exponential: Method 2 . . . . . . . . . . . . 1076.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4 Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.5 Prospects for Physical Point Calculation . . . . . . . . . . . . . . . . . . . 114

6.5.1 Simulation with 3 flavours . . . . . . . . . . . . . . . . . . . . . . . 1156.5.2 3 Flavour Renormalisation . . . . . . . . . . . . . . . . . . . . . . . 116

7 Results of K → πνν̄ Simulations 1197.1 Details of the Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207.2 Setup of the Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2.1 Details of the Implementation . . . . . . . . . . . . . . . . . . . . . 1227.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.3.1 2pt and 3pt Correlators . . . . . . . . . . . . . . . . . . . . . . . . 1247.3.2 Z-Exchange Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.3.2.1 Vector Current . . . . . . . . . . . . . . . . . . . . . . . . 1297.3.2.2 Axial Current . . . . . . . . . . . . . . . . . . . . . . . . 1337.3.2.3 Disconnected Diagrams . . . . . . . . . . . . . . . . . . . 134

7.3.3 W-W Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.3.3.1 Type 1 Diagrams . . . . . . . . . . . . . . . . . . . . . . . 1347.3.3.2 Type 2 Diagrams . . . . . . . . . . . . . . . . . . . . . . . 139

7.4 Renormalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.4.1 RI-SMOM renormalisation . . . . . . . . . . . . . . . . . . . . . . . 1447.4.2 Perturbative Matching . . . . . . . . . . . . . . . . . . . . . . . . . 1477.4.3 Final Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8 Conclusions 153

A Approximations 155A.1 c20(k) = −c10(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155A.2 SU(3) symmetric limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

B Perturbative Results 157B.1 Expressions for ∆YAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

List of Figures

2.1 Diagrams giving rise to Q1 and Q2 (current-current) operators. . . . . . . 202.2 Example diagrams contributing to the calculation of Zij . . . . . . . . . . . 21

3.1 Penguin diagrams that give rise to (a) Q3,..,6 and (b) Q7,...,10. . . . . . . . 283.2 Penguin diagrams that gives contributions to the effective s→ d`¯̀ vertex

in rare kaon decays. When ` = ν, there is no photon penguin diagram. . 313.3 Box diagrams that give contributions to the effective s→ d`¯̀ vertex. . . . 313.4 Diagrams associated with (a) electromagnetic and (b) chromomagnetic

penguin operators respectively. The cross indicates a mass insertion. . . . 323.5 (a) Diagrams that give rise to Q∆S=1`q (s) and Q

∆S=0`q (d), resulting in the

local operators shown in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . 353.6 The one-loop contribution to the decays K → πγ∗ arising as ππ → γ∗

rescattering in K → πππ decays. . . . . . . . . . . . . . . . . . . . . . . . 373.7 The contributions to KL → π`+`− decays from (a) indirect CP -violation

via kaon oscillation and (b) a CP -conserving 2-photon exchange. . . . . . 38

4.1 Diagram that contributes to the 2pt pion correlation function Eq. (4.58). . 554.2 Diagrams that result from Wick contractions of the 3pt correlation func-

tion with a current insertion Eq. (4.70). The double line shows the prop-agator that may be obtained as a result of a sequential inversion. . . . . . 58

4.3 Diagrams that result from Wick contractions of the 3pt correlation func-tion with a 4-quark weak operator (Qi) insertion Eq. (4.77). . . . . . . . . 59

4.4 Kaon correlator, fit to the ansatz Eq. (4.99). (a) shows the effectivemass plot used to determine the region where the ground state dominates,highlighted by the horizontal line; (b) shows the fit to the folded correlator.Errors have been computed using bootstrap resampling. . . . . . . . . . . 64

5.1 The four diagram topologies obtained after performing the Wick contrac-tions of the charged pion and kaon interpolating operators with the HWoperator. The light quark flavour indicated by u/d is u for charged mesonsand d for neutral mesons. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 The five possible current insertions for the C class of diagrams. The lightquark flavour indicated by u/d is u for the charged decays and d for theneutral decay. The fifth diagram shown is a quark-disconnected topology. 72

5.3 The additional two classes of diagrams obtained after performing the Wickcontractions of the neutral pion and kaon interpolating operators with theHW operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.4 The two classes of diagrams the must be computed for W-W exchangeamplitudes. The internal lepton propagator will correspond to ` = e, µ, τ . 73

ix

x LIST OF FIGURES

5.5 Schematic of the momentum flow for the renormalisation condition of a2pt Green’s function using momentum subtraction schemes. For RI-MOMschemes we require p1 = p2, p2i = µ

2RI; for RI-SMOM schemes we require

p21 = p21 = (p1 − p2)2 = µ2RI. . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.6 Schematic of the momentum flow for the renormalisation condition of a4pt Green’s function using the RI-SMOM scheme. In this scheme themomenta satisfy p2i = (p1 − p2)2 = (p3 − p4)2 = µ2RI. . . . . . . . . . . . . 88

5.7 Diagrams leading to UV divergencs in (a) Z- and γ-exchange diagramsand (b) W-W diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1 Demonstration of how propagators are used to construct diagrams. Theposition of theHW operator is indicated by the shaded square, and may beplaced at any spacetime position. The insertion of the current is denotedby a black square, fixed on an single time slice and summed over space.The double line represents the part of the propagator computed usinga sequential inversion; the dotted line represents the loop propagator,computed using spin-color diluted random volume sources. . . . . . . . . . 97

6.2 Plots showing fits to (a) kaon and (b) pion (folded) correlators. The sourcetimes for each meson are shown by the vertical black lines. . . . . . . . . . 101

6.3 Plots of fits to 3pt correlation functions used to extract HW matrix ele-ments. Results for p = (1, 1, 0) and p = (1, 1, 1) are not shown as theyare too noisy to extract a significant signal. . . . . . . . . . . . . . . . . . 102

6.4 Plots of fits to 3pt (a) kaon and (b) pion correlation functions with avector current insertion. In each case the initial meson is at rest; thelegend indicates the momentum of the final state meson. . . . . . . . . . . 103

6.5 Determination of the parameter cs from a fit to the ratio of 3pt HWand s̄d correlators. The corresponding ratio of the 4pt correlator (withpπ =

2πL (1, 0, 0)) is also shown. The position of the plateau corresponds

to cs = 0.000240(8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.6 The contributions of each of the diagrams to the rare kaon decay corre-

sponding to the weak operators (a) Q1 and (b) Q2, both before and afterthe GIM subtraction, shown for the example kinematic of pK = (0, 0, 0),pπ =

2πL (1, 0, 0). Each diagram has been constructed using the appropri-

ate fractional quark charges (excluding the overall charge factor e), and thecorrelators have been multiplied by the relevant renormalisation constantsand Wilson coefficients for matching to the MS scheme. Time positionsof the kaon/pion interpolators and current insertion are indicated. . . . . . 104

6.7 (a) The 4pt rare kaon decay correlator measured in our simulation withk = (0, 0, 0) and p = 2πL (1, 0, 0). The ground state contribution has beenconstructed from fits to 2pt and 3pt correlators. (b) The 4pt correlator af-ter removing the ground state contribution (i.e. the single-pion and singlekaon intermediate states). Time positions of the kaon/pion interpolatorsand the current insertion are indicated. . . . . . . . . . . . . . . . . . . . . 105

6.8 Plot of the amplitudes (in lattice units) obtained using each of the differentanalysis methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

LIST OF FIGURES xi

6.9 The integrated 4pt correlator, shown for (a) Tb fixed at its lower limitto demonstrate the Ta dependence and (b) Ta fixed at its lower limitto demonstrate the Tb dependence. The kinematics shown are for pπ =(1, 0, 0), pπ = (1, 1, 0), pπ = (1, 1, 1) top to bottom. The single-pionexponential growth has been removed using method 1, with the approxi-mationMH (pπ) = MH (pK). The position of the plateaus correspondsto A0 = −0.0028(6), A0 = −0.0028(18), A0 = −0.0050(38) top to bottom,obtained by fits to the data over the indicated ranges. . . . . . . . . . . . 108

6.10 The integrated 4pt correlator, shown for (a) Tb fixed at its lower limitto demonstrate the Ta dependence and (b) Ta fixed at its lower limitto demonstrate the Tb dependence. The kinematics shown are for pπ =(1, 0, 0), pπ = (1, 1, 0), pπ = (1, 1, 1) top to bottom. The single-pionexponential growth has been removed using method 2. The position ofthe plateaus corresponds to A0 = −0.0027(6), A0 = −0.0028(18), A0 =−0.0053(39) top to bottom, obtained by fits to the data over the indicatedranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.11 The integrated 4pt correlator, shown for (a) Tb fixed at its lower limitto demonstrate the Ta dependence and (b) Ta fixed at its lower limitto demonstrate the Tb dependence. The kinematics shown are for pπ =(1, 0, 0), pπ = (1, 1, 0), pπ = (1, 1, 1) top to bottom. The single-pionexponential growth has been removed using method 1, with the approxi-mationMsd (pπ) =Msd (pK). The position of the plateaus correspondsto As̄d0 = −0.00001(8), As̄d0 = −0.00002(21), As̄d0 = 0.00032(52) top tobottom, obtained by fits to the data over the indicated ranges. . . . . . . 111

6.12 Plot of the amplitudes (in lattice units) obtained using each of the differentanalysis methods for the C(4)s̄d correlator. The expected value of zero ismarked explicitly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.13 Dependence of the form factor for the decay K+ → π+`+`− upon z =q2/M2K . Our lattice data is fit to a linear ansatz to obtain a = 1.6(7) andb = 0.7(8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.14 Green’s functions that must be computed for the renormalisation proce-dure, corresponding to (a) E and (b) S loop contractions. The exactgamma matrix insertion will depend upon the weak operator Qi. In (c)we show the Green’s function required for calculation of the counterterm. 117

7.1 Dalitz plot for K → πνν̄ (with physical masses) showing the physicalkinematical region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.2 Plots of (a) kaon and (b) pion 2pt correlation functions with wall sourceand sink smearing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.3 Plots of (a) kaon and (b) pion 2pt correlation functions with a wall sourceand either a point sink or a local axial current sink. . . . . . . . . . . . . . 126

7.4 Plot of the determination of the parameter cs for the operators Q1 andQ2 that make up HW . We obtain c1s = 7.7(3) × 10−5 for Q1 and c2s =1.86(3)× 10−4 for Q2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.5 Plots of fits to K`3 correlators (with Lorentz index in time direction, µ =0), for (a) q2 = q2max (initial and final state at rest) and (b) q2 = 0, afterdividing out source/sink factors and ground state time dependence. Thehorizontal lines indicate the best fits and indicate the fit ranges. . . . . . . 127

xii LIST OF FIGURES

7.6 Plot of fit to pion correlator with a local vector current insertion (withLorentz index in time direction, µ = 0), after dividing out source/sinkfactors and ground state time dependence. The horizontal line indicatesthe best fit and indicates the fit range. . . . . . . . . . . . . . . . . . . . . 127

7.7 Plot of fit to K`3 correlators, with spatial Lorentz indices, after dividingout source/sink factors and ground state time dependence. The threespatial directions are fit simultaneously with a single parameter. Thehorizontal line shows the best fit result and indicates fit range. . . . . . . 128

7.8 Plots of the unintegrated 4pt correlator for Z-exchange diagrams witha vector current insertion, for each operator that makes up HW . Thecorrelator is shown (a) before and (b) after shifting by the scalar densitys̄d to remove the growing exponential term of the single-pion intermediatestate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.9 The integrated 4pt correlator for Z-exchange diagrams with a vector cur-rent insertion, shown with (a) varying Ta and fixed Tb and (b) vice versa,with pK = pπ = 0. Results are shown for the operators Q1 (top) and Q2(bottom). The result for the matrix element is indicated by the horizontalband and indicates the fit range used for its extraction. . . . . . . . . . . . 131

7.10 The integrated 4pt correlator for Z-exchange diagrams with a vector cur-rent insertion, shown with (a) varying Ta and fixed Tb and (b) vice versa.Results are shown for the operators Q1 (top) and Q2 (bottom). The resultfor the matrix element is indicated by the horizontal band and indicatesthe fit range used for its extraction. . . . . . . . . . . . . . . . . . . . . . . 132

7.11 Plots of the unintegrated 4pt correlator for Z-exchange diagrams with anaxial current insertion, for each operator (a) Q1 and (b) Q2 that makesup HW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.12 The integrated 4pt correlator for Z-exchange diagrams with an axial cur-rent insertion (µ = 0), shown with (a) varying Ta and fixed Tb and (b)vice versa. Results are shown for the operators Q1 (top) and Q2 (bottom).The result for the matrix element is indicated by the horizontal band andindicates the fit range used for its extraction. . . . . . . . . . . . . . . . . 135

7.13 The integrated 4pt correlator for Z-exchange diagrams with an axial cur-rent insertion (µ = i), shown with (a) varying Ta and fixed Tb and (b)vice versa. Results are shown for the operators Q1 (top) and Q2 (bot-tom). The result for the matrix element is indicated by the horizontalband and indicates the fit range used for its extraction. . . . . . . . . . . . 136

7.14 Plots of the unintegrated 4pt correlator for quark-disconnected Z-exchangediagrams with an axial current insertion, for each operator that makes upHW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.15 Type 1 unintegrated correlators FWW for the three different flavours ofintermediate leptons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.16 Unintegrated 4pt correlator for W-W diagram with intermediate muon,after removing the µ+ν intermediate state. The horizontal black line showsthe position of zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.17 The integrated 4pt correlator for type 1 W-W diagrams, shown for (a) Tbfixed to demonstrate the Ta dependence and (b) vice versa. Integrationlimits and intermediate lepton flavour are indicated within the legend.For the electron and muon I show the Tb dependence before and after itsremoval using a direct fit of the 4pt correlator. . . . . . . . . . . . . . . . 140

LIST OF FIGURES xiii

7.18 Type 2 unintegrated correlators FWW for the three different leptonic in-termediate states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.19 The integrated 4pt correlator for type 2 diagrams, shown for (a) Tb fixedto demonstrate the Ta dependence and (b) vice versa. Integration limitsare indicated within the legend. For the muon I show the Ta dependencebefore and after removing the π0`+ν intermediate state exponential. . . . 142

7.20 Results for the ratio R [Eq. (7.18)], i.e. the form factors obtained frombilocal matrix elements, normalised by the K`3 form factor f+

(q2), for

W-W diagrams (left) and Z-exchange diagrams (right). The horizontalbands shows the results before the regulation of the divergence in the RI-SMOM scheme; the points show the RI-SMOM regulated results. Theseresults additionally include the normalisation factor π2/λ4M2W . . . . . . . 146

7.21 Diagrams that must be computed for the perturbative matching of (a)Z-exchange and (b) W-W diagrams. . . . . . . . . . . . . . . . . . . . . . 147

7.22 Dependence of the quantities YAB (µ, µRI), rAB (µ, µRI) and ∆YAB (µ, µRI)on the renormalisation scale µ = µRI. Results for the W-W diagram areshown on the left, and Z-exchange on the right (matching is only requiredfor the axial current). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.23 Plot summarising the total results for the ratio Eq. (7.18) for K → πνν̄decays. On the left I show the results regulated on the lattice (horizontalband), in the RI-SMOM scheme (blue circles) and in the MS scheme (greentriangles). On the right I show the difference between the results obtainedfrom the lattice and the results obtained using perturbation theory. Notehow the discrepancy between the lattice and perturbative results is largefor both WW and Z-exchange diagrams, but these discrepancies largelycancel in the total result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

List of Tables

2.1 Matter content of the Standard Model, along with representations andcharges under the Standard Model gauge group SU (3)× SU (2)× U (1).The index i on the fermion fields runs over the 3 generations. . . . . . . . 9

2.2 Quantum numbers of Dirac bilinears of the form ψ̄Γψ. . . . . . . . . . . . 11

5.1 Branching ratios and decays widths relevant to rare kaon decays. Thedecays with (semi-)leptonic final states are relevant for intermediate statesin W-W diagrams; those with pure hadronic final states are relevant forintermediate states in Z- and γ-exchange diagrams. . . . . . . . . . . . . . 80

5.2 Results for the MS Wilson coefficients for HW [Eq. (5.2)], the MS →RI matching, the RI renormalisation matrix and the final lattice Wilsoncoefficients, all computed using µ = µRI = 2.15 GeV [1]. . . . . . . . . . . 90

6.1 Summary of propagators calculated in our simulation for a single choice ofpion momentum on a single configuration, and the corresponding numberof inversions required. Nη is the number of noise vectors used in thecomputation of the quark loops; Nt is the number of translations in thetime direction across a single configuration at which all the contractionsare computed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2 Table showing fit results to 2pt and 3pt correlators. The double lineseparates results obtained from 2pt (above) and 3pt (below) correlators. . 101

6.3 Parameters of Eq. (6.8) (in lattice units) obtained via analytic reconstruc-tion using 2pt and 3pt fit results or fitting the integrated 4pt correlatordirectly. For c20 the result using the approximationMH (pπ) =MH (pK)is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4 Summary of matrix elements obtained using various analysis methods.All values are given in lattice units. Results are shown for all classes ofdiagrams, and also separated into the nonloop and loop contributions. . . 109

6.5 The form factor of the K (0) → π (pπ) γ∗ decay computed for the threepion momenta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.1 Summary of propagators calculated in our simulation on a single configu-ration, and the corresponding number of inversions required. In the lastline I give the total number of inversions including both periodic (P) andanti-periodic (AP) boundary conditions. Free lepton propagators thatdon’t require an inversion are not included here. T is the time extent ofthe lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.2 Table of fit results of 2pt correlation functions involving a pion (P = π)and kaon (P = K), with pπ = −0.0414(1, 1, 1). . . . . . . . . . . . . . . . 125

xv

xvi LIST OF TABLES

7.3 Summary of fit results for 3pt matrix elements (denoted byM) and corre-sponding form factors required for the analysis of K → πνν̄ decays. In thethird row of results the final entry corresponds to the pion electromagneticform factor, Fπ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.4 Summary of fit results for Z-exchange analysis. The double line separatesresults for the vector current (above) from the results for the axial current(below). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.5 Summary of form factors obtained from Z-exchange analysis. The uppercase F denotes the Z-exchange form factor, obtained using Eqs. (5.6)and (5.10). The lower case f denotes the K`3 form factor, obtained usingEq. (5.24). The double line separates results for the vector current (above)from the results for the axial current (below). . . . . . . . . . . . . . . . . 129

7.6 Counterterms for removing the short-distance divergent in Z-exchangediagrams with a vector current insertion. . . . . . . . . . . . . . . . . . . . 133

7.7 Fit results [to Eq. (7.13)] for the Kπ`+ν intermediate states in type 1W-W diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.8 Summary of fit results for W-W analysis. The results are shown beforemultiplying by the relevant individual operator renormalisation constants. 143

7.9 Counterterms for removing the short-distance divergence in WW diagramsand Z-exchange diagrams with an axial current insertion. The countert-erms are calculated separately for each operator Qi entering HW for Z-exchange diagrams, and for each internal lepton in W-W diagrams. Re-sults are quoted in units of 10−2. . . . . . . . . . . . . . . . . . . . . . . . 145

7.10 Summary of input parameters and matching scales used for results ob-tained using RG-improved perturbation theory. . . . . . . . . . . . . . . . 146

Declaration of Authorship

I, Andrew J. Lawson , declare that the thesis entitled Exploratory Lattice QCD Studiesof Rare Kaon Decays and the work presented in the thesis are both my own, and havebeen generated by me as the result of my own original research. I confirm that:

• this work was done wholly or mainly while in candidature for a research degree atthis University;

• where any part of this thesis has previously been submitted for a degree or anyother qualification at this University or any other institution, this has been clearlystated;

• where I have consulted the published work of others, this is always clearly at-tributed;

• where I have quoted from the work of others, the source is always given. With theexception of such quotations, this thesis is entirely my own work;

• I have acknowledged all main sources of help;

• where the thesis is based on work done by myself jointly with others, I have madeclear exactly what was done by others and what I have contributed myself;

• parts of this work have been published as: [2] and [3]

Signed:.......................................................................................................................

Date:..........................................................................................................................

xvii


Acknowledgements

I would first like to thank my supervisors, Prof. Chris Sachrajda and Dr. Andreas Jüttner,for their support and guidance throughout this project.

Secondly I extend my gratitude to Dr. Antonin Portelli, who provided me with muchsupport and advice, both on computational and theoretical issues surrounding the rarekaon decays K → π`+`−. I am also indebted to Xu Feng, whose careful advice wasinstrumental in developing my understanding of the analysis and renormalisation ofK → πνν̄ decays.

I’d also like to thank my colleagues from the RBC-UKQCD collaboration for their usefulinput. Special thanks go to Prof. Norman Christ for his excellent insights into the rarekaon decay projects and to Prof. Peter Boyle, whose computational efforts underpin largeamounts of this work.

I would also like to extend my thanks to my fellow 4007 office members, who have madethese last 4 years very memorable. I am grateful to my parents for their constant loveand support. Finally, I would like to thank Madi for putting up with me while I workedon strange physics.

xix

Chapter 1

Introduction

The rare kaon decays K → π`+`− and K → πνν̄ proceed via flavour-changing neutralcurrents (FCNCs), i.e. s→ d transitions, and hence they first arise only as second-orderelectroweak processes. This suppression makes these decays ideal for probing for NewPhysics, by searching for discrepancies between Standard Model predictions of theseprocesses and experimental results. The study of rare kaon decays has thus attractedincreasing interest in recent years.

The phenomenology behind each of the rare kaon decays K → π`+`− and K → πνν̄ ismarkedly different [4]. The decays K → πνν̄ are known to be short-distance dominated,and hence have traditionally been theoretically cleaner to study thanK → π`+`− decays,which see large contributions from long-distance, hadronic effects. To be more specific,studies of rare kaon decays are usually performed in the context of a low-energy effectivetheory, where the heavy weak bosons and heavy quarks do not appear as dynamicaldegrees of freedom. The low-energy (long-distance) hadronic effects are thus separatedfrom the high energy (short-distance) physics; the latter contributions may be computedusing perturbation theory.

In K → πνν̄ decays, the presence of a quadratic GIM mechanism [5] enhances thecontributions of heavy quark loops; as a result these decays are short-distance dom-inated. The required hadronic contributions may be obtained from measurements ofsemileptonic K+ → π0`+ν decays. For the decay KL → π0νν̄, the dominant contri-bution originates from the direct CP -violating amplitude, hence is proportional to theCabibbo-Kobayashi-Maskawa (CKM) matrix factor Im (λq) (where λq = V ∗sqVqd), whichsignificantly suppresses the up and charm contributions. As a result, this decay is en-tirely dominated by loops involving the top quark, and is thus the theoretically cleanestrare kaon decay channel. A recent theoretical prediction for this branching ratio is [6]

Br(KL → π0νν̄

)= 3.00(30)× 10−11, (1.1)

1

2 Chapter 1 Introduction

where the majority of the error originates from uncertainties in input SM parameters. Forthe CP -conserving decay K+ → π+νν̄, a recent theoretical prediction for the branchingratio is [6]

Br(K+ → π+νν̄

)= 9.11(72)× 10−11. (1.2)

The majority of the error again originates from parametric uncertainties in StandardModel parameters; however long-distance contributions of the charm (and up) quarksalso represent significant sources of error. It will be important to constrain this sourceof error as experimental measurements of Br (K+ → π+νν̄) improve. Hence an opportu-nity for lattice QCD is to provide estimates of the long-distance contributions to theseamplitudes, such that the uncertainty from long-distance contributions may be reduced.

On the experimental side, K+ → π+νν̄ is challenging to measure. The current experi-mental estimate for the branching ratio is [7]

Br(K+ → π+νν̄) = 1.73+1.15−1.05 × 10−10, (1.3)

based on results collected by the experiments E787 [8–11] and E949 [7, 12] at BNL. Thenew NA62 experiment at CERN [13, 14] is currently aiming to measure approximately 80K+ → π+νν̄ events over a period of two years, thus reducing the error on the branchingratio to around 10%. Data acquisition began in 2016, and an analysis of 1012 kaon decayscollected so far is underway [15]. The initial goal of NA62 was to reduce the error onthe CKM matrix parameter |Vtd|; however additional physics goals include measuringBr (K± → π±`+`−) to greater accuracy than present, and putting constraints on leptonflavour violating decays such as K+ → π+µ+e−.

The decays KL → π0νν̄ are particularly challenging to measure, given that all particlesin the final state are neutral. The E391a experiment at KEK [16] previously set an upperbound for the branching ratio at

Br(KL → π0νν̄

)≤ 2.6× 10−8 at 90% confidence level. (1.4)

At present there exists a dedicated experiment at J-PARC (KOTO) [17] to measurethe KL → π0νν̄ branching ratio. The KOTO experiment has reported one candidateKL → π0νν̄ event so far based on an analysis of data collected in 2013 [18]. It isworth noting that there is no current plan to measure the decay KS → π0νν̄, which isprohibitively difficult to detect.

On the other hand, the CP -conserving processes KS → π0`+`− and K+ → π+`+`−, me-diated predominantly via a single-photon exchange, are long-distance dominated [2, 19].The reason for this is that the short-distance top quark contribution is suppressed bythe CKM factor Re (λt) and even a potentially large light-quark short-distance con-tribution is cut off at the charm quark Compton wave length by a logarithmic GIM

Chapter 1 Introduction 3

cancellation [2, 5]. Previous studies of these decays have been performed using chi-ral perturbation theory [20–23]. KL → π0`+`− decays however do have a significantshort-distance component proportional to the CP -violating CKM parameter Im (λq).However they also contain an indirect CP -violating contribution due to neutral kaonoscillation [24]. There is significant interference between these two contributions; how-ever it is not possible to determine from experiment whether the inteference is positiveor negative. Because of the dominance of long-distance contributions in the decaysKS → π0`+`− and K+ → π+`+`−, there are no current estimates for the branchingratios of these processes from first principles. This hence motivates a lattice calculationof these amplitudes [19, 25].

On the experimental side, the branching ratios for K+ → π+`+`− processes are knownto a high degree of accuracy [26, 27]:

Br(K+ → π+e+e−

)= 3.14(10)× 10−7, (1.5)

Br(K+ → π+µ+µ−

)= 9.62(25)× 10−8, (1.6)

which were measured at the CERN NA48/2 experiment. As NA62 increases the sta-tistical precision of these branching ratios, the experiment may become sensitive tolepton flavour universality violation in rare kaon decays [28]. KS → π0`+`− decayshowever are more challenging to measure, although their detection is important for cal-culating the indirect CP -violating contribution to KL → π0`+`− decays via the chainKL → K1 → π0`+`−, where K1 is the CP -even component of KL. The branching ratiosare currently only known with ∼ 50% errors [29, 30]:

Br(KS → π0e+e−

)=(5.8+2.9−2.4

)× 10−9, (1.7)

Br(KS → π0µ+µ−

)=(2.9+1.5−1.2

)× 10−9. (1.8)

Given the difficulty of the experimental measurement, there exists a good opportunityto extract this result instead from lattice QCD simulations. In addition, such a latticecalculation would determine the phase describing the interference between the indirectand direct CP -violating amplitudes, which cannot be determined from experimentalmeasurements of KS → π0`+`− branching ratios. Neither NA62 nor KOTO aim tomeasure this decay, however LHCb are currently exploring the prospect of studying rareKS decays after the next shutdown and upgrade [31].

In this thesis I will report on our exploratory studies of the rare kaon decays K →π`+`− [2] and K → πνν̄ [3, 32]. The main objective of these studies is to demonstratethe theoretical techniques of Refs. [19, 25, 33], such that the desired rare kaon decaymatrix elements may be extracted from lattice QCD simulations. Secondly we can usethese exploratory studies to evaluate the feasibility of a physical-point determination ofthe long-distance contributions to rare kaon decay amplitudes, such that comparisonsmay ultimately be made with experimental data.

4 Chapter 1 Introduction

The layout of this thesis is as follows. Chapters 2, 3 and 4 present much of the theoreticalbackground required for this thesis. In chapter 2 we start with an introduction to theStandard Model, where I introduce many of the fundamental concepts required for thisthesis. I begin in section 2.1 with a discussion of QCD and the weak interaction, anddiscuss their symmetries. Following this I introduce Fermi effective theory in section 2.2,which is required to study low-energy processes mediated by the weak interaction. Sub-sequently in section 2.3 I will introduce chiral perturbation theory, which has been usedin previous studies of the long-distance contributions to rare kaon decays. In chapter 3I provide a review of the current theoretical understanding of rare kaon decays, buildingon the discussions presented above. I will introduce the effective Hamiltonians requiredto study each decay at low energies in section 3.1. In section 3.2 I go on to review thecurrent theoretical understanding of these decays, which so far has not included latticeQCD. Chapter 4 then introduces lattice QCD, where I will introduce the concepts behindthe discretisation of QCD in section 4.1, as well as details of the numerical simulation insection 4.2. I then move onto an explanation of some of the technical details regardingthe construction of correlation functions from our simulation data in section 4.3. I followthis by presenting a generic discussion of the analysis of correlation functions obtainedfrom numerical simulations in section 4.4.

Chapter 5 then introduces the concepts required to study rare kaon decays using latticeQCD. I start in section 5.1 by showing how the operators introduced in chapter 3 translateinto correlation functions that may be studied. The analysis of these correlation functionsis then introduced in section 5.2. In this section I demonstrate a generic feature ofthe evaluation of four-point (4pt) correlators in Euclidean spacetime: the presence ofintermediate states lighter than the initial particle state give rise to contributions whichgrow exponentially with the separation of the two operators involved in the 4pt correlator.I hence discuss the exponentially growing contributions that must be removed from ourrare kaon decay correlators in order to obtain the desired matrix elements. In section 5.3I discuss the renormalisation of each individual operator entering the 4pt correlators, aswell as the regulation and removal of additional short-distance divergences caused by thecontact of the two operators in the 4pt correlators.

At this point the theoretical stage is set, and I move on to a discussion of the results ofour rare kaon decay simulations. In chapter 6 I begin with a discussion of our simulationsof K → π`+`− amplitudes, which are conceptually simpler as short-distance divergencesarising in our lattice simulation cancel automatically via the GIM mechanism. Thesimulation details and setup are discussed in sections 6.1 and 6.2 respectively. I thenmove onto a discussion of the numerical results in section 6.3, where I present a detailedanalysis of the 4pt correlators. In section 6.4 I present a demonstrative calculation toshow how our results might be compared to existing predictions of chiral perturbationtheory. I then discuss the future plans for a physical point simulation in section 6.5.In chapter 7 I discuss the results of our K → πνν̄ simulations, where there is a short

Chapter 1 Introduction 5

distance divergence that does not automatically cancel on the lattice. The regulation ofthis divergence must hence be converted into a continuum scheme. For this we choosethe MS scheme, such that we can combine the lattice result with existing results for theshort-distance contributions to K → πνν̄ decays to obtain a finite result. The simulationdetails and setup are discussed in sections 7.1 and 7.2 respectively. The analysis of thelattice results is discussed in detail in section 7.3. In section 7.4 I present the stepsrequired to match the lattice-regulated short-distance divergences in the 4pt correlatorsto the continuum MS scheme, such that the final answer combining both long-distancelattice results and short-distance perturbative results is finite. Finally in chapter 8 Ipresent my conclusions.

Chapter 2

Standard Model

The Standard Model of particle physics describes three of the four fundamental forcesof nature: electromagnetism, and the weak and strong interactions; only gravity is miss-ing. The theory is remarkably robust: barring the detection of neutrino masses, nonew physics beyond the Standard Model has been confirmed since its inception. It isclear however that the theory requires extension, as there are many observed phenom-ena (besides gravity) that it does not explain. For example, the Standard Model doesnot contain a good dark matter candidate, which would explain the observed velocitydistribution in galaxy rotation curves [34]. Additionally the amount of CP -violation inthe Standard Model is not enough to satisfy the Sakharov conditions [35], necessary toproduce the observed matter-antimatter asymmetry in the universe.

The Standard Model does however appear to be under direct strain in certain areas.There are many ongoing experimental efforts to make precision tests of the StandardModel in the hope of identifying the need for new physics, with the measurement of theanomalous magnetic moment of the muon (g− 2) being a particularly well-known exam-ple [36]. The current discrepancy between theory and experiment stands at ∼ 3.5σ [37].Experiments at Fermilab [38] and J-PARC [39] aim to increase the experimental preci-sion further, and on the theoretical side there are many efforts to reduce the hadronicuncertainties using lattice QCD [40–47].

Tensions also exist in flavour physics: many experimental results for B-meson decaysshow small deviations from the Standard Model. For example, LHCb has detected 3.5σdiscrepancies in Bs → φµ+µ− decays [48]. Further promising deviations have been foundby in B → Dτντ and B → D∗τντ decays, where there is a combined discrepancy of ∼ 4σbetween Belle [49], BaBar [50] and LHCb [51] experiments.

In order to support the experimental efforts, it is important to be able to make accu-rate theoretical predictions using the Standard Model, such that any deviations may bequantified. In this chapter I will hence provide an introduction to the Standard Model,

7

8 Chapter 2 Standard Model

beginning with a cursory overview in section 2.1. This section will introduce standardtextbook definitions; books such as Ref. [52] may be consulted for further details. Oncethe scene is set, I will introduce the details for more specified areas of physics requiredfor this thesis. First I introduce the effective theory used to describe the weak interactionin section 2.2, which is required to analyse weak matrix elements involved in rare kaondecays at low scales O (1 GeV) that are within reach of a lattice simulation. SecondlyI will introduce chiral perturbation theory in section 2.3, which is a tool to analyse thelow-energy dynamics of pseudoscalar mesons. This theory has been previously appliedto study the rare kaon decays K → π`+`− that are dominated by low-energy hadroniceffects, and to estimate the small long-distance corrections to K → πνν̄ amplitudes.

2.1 Particle Content and Interactions

The gauge group of the Standard Model is SU (3)×SU (2)×U (1), with particle contentas shown in Table 2.1. The gauge group encodes the fundamental forces of nature thatthe Standard Model describes. SU(3) describes the strong force, QCD, and SU(2)×U(1)describes the electroweak interaction, which is broken to the subgroup U(1) below theweak scale; this residual gauge symmetry describes electromagnetism. The spontaneoussymmetry breaking of SU(2)×U(1)→ U(1) is described by the Higgs mechanism, whichwe will touch on briefly in section 2.1.2.1.

The matter content of the Standard Model is as follows. Firstly we have the fermionsector, of which we define two types: quarks and leptons. The quark sector is made upof three generations of “up-type” quarks (up, charm and top), and three generations of“down-type” quarks (down, strange and bottom). Quarks are the only particles in theStandard Model that interact via SU(3); they additionally interact via the weak force andelectromagnetism. The lepton sector is made up of three generations of leptons carryingelectromagnetic charge (electron, muon, tau), along with corresponding neutrinos, whichdo not interact electromagnetically. Secondly we have the gauge sector: the stronginteraction is mediated by gluons; the weak interaction by the W and Z bosons, andelectromagnetism by the photon. The gluon and photon are massless, as their gaugeinteractions are unbroken in the Standard Model. Finally we have the scalar (or Higgs)sector, which contains only a single spin-0 particle: the Higgs boson. The W and Zbosons, leptons and quarks all acquire masses at low energies when the Higgs bosonacquires a non-zero vacuum expectation value (VEV).

In the following section I will describe the various elements of the Standard Model inmore detail which are central to this thesis. I will begin by discussing QCD and itssymmetries, which are important preliminaries before we discuss the its discretisation inchapter 4. I will then go on to describe the electroweak interaction in more detail, whichis responsible for the decays being studied in this thesis.

Chapter 2 Standard Model 9

Name SU (3) SU (2) U (1) Spin

Quarks (×3 generations)

Qi 3 21

6

1

2

ui 3 1 −2

3

1

2

di 3 1 −1

3

1

2

Leptons (×3 generations)Li 1 2 −

1

2

1

2

ei 1 1 11

2

Gluon G 8 1 0 1

Electroweak bosonsW 1 3 0 1

B 1 1 0 1

Higgs H 1 2 −12

0

Table 2.1: Matter content of the Standard Model, along with representationsand charges under the Standard Model gauge group SU (3) × SU (2) × U (1).The index i on the fermion fields runs over the 3 generations.

2.1.1 QCD

QCD is the theory of the SU(3) gauge interaction between quarks and gluons. TheLagrangian of QCD is given by

LQCD = −1

4GaµνG

aµν + qf (i��D −mf ) qf , (2.1)

where in addition to Lorentz indices (greek indices), the index f runs over the six flavoursof quark, qf , and the index a = 1, ..., 8 runs over colour, i.e. an index for each generator ofSU (3). I have suppressed the colour and Dirac indices of the quark fields. The covariantderivative is given by

Dµ = ∂µ + igTaAaµ, (2.2)

where T a are the generators of the SU (3) algebra and g is the QCD coupling. The gluonfield strength tensor, Gaµν , is defined by

Gaµν = ∂µAaν − ∂νAaµ + igfabcAbµAcν , (2.3)

where Aaµ is the gauge field of the gluon and fabc are the SU(3) structure constants [52].


2.1.1.1 Discrete Symmetries

Besides the SU(3) gauge symmetry, there are global symmetries of QCD, which will beuseful for lattice studies. QCD exhibits three discrete symmetries: charge conjugation(C), parity (P ) and time-reversal (T ). Each of these transformations may be written asunitary operators, transforming a field ψ → UψU †. These symmetries are particularlyimportant for working out the transformation properties of Dirac bilinears. I will hencesummarise the effects of these symmetries below [52].

Charge conjugation involves transforming a particle into one with the opposite charge,i.e. its anti-particle. The momentum and spin of the particle are unchanged. The fermionfields transform as

ψ (x)→ −i(ψ (x) γ0γ2

)T, (2.4)

ψ (x)→ −i(γ0γ2ψ (x)

)T. (2.5)

Fermions are not eigenstates of charge conjugation (e.g. charge conjugation changesquark flavour); however we may construct Dirac bilinears that are indeed C eigenstates.

A parity transformation encodes a spacial reflection of a particle, i.e. transformingx→ xP , with x = (xt,x) and xP = (xt,−x). Parity therefore reverses the momentum ofa particle, but does not affect the spin. Under such a transformation, the fermion fieldstransform as

ψ (x)→ ηaγ0ψ (xP ) , (2.6)ψ (x)→ η∗aψ (xP ) γ0, (2.7)

where ηa is a complex phase. Finally we have time-reversal, which is defined somewhatanalogously to parity as x → xT , with xT = (−xt,x). Time-reversal flips both the spinand momentum of a particle. The fermion fields transform as

ψ (x)→ γ1γ3ψ (xT ) . (2.8)ψ (x)→ −ψ (xT ) γ1γ3. (2.9)

Strictly speaking, time-reversal is an antiunitary operator. For the above transformationsto hold, T must also act on c-numbers c as Tc = c∗T . For example, T acts on the time-evolution operator eiHt as TeiHt = e−iHtT , effectively changing the sign of t [52].

From the transformations given above we see that the fermion fields are not eigenstatesof these symmetries; however Dirac bilinears constructed from these fields are. Table 2.2gives a summary of the transformation of Dirac bilinears under the transformations Cand P . This will become particularly relevant when we consider the creation of QCDbound states. A pion for example is a pseudo-scalar particle, as such we may create a


State JPC Γ

Scalar 0++ 1, γ0

Pseudo-scalar 0−+ γ5, γ0γ5

Vector 1−− γi, γ0γi

Axial-vector 1++ γiγ5

Tensor 1+− γiγj

Table 2.2: Quantum numbers of Dirac bilinears of the form ψ̄Γψ.

(positively-charged) pseudo-scalar state using

u (x) γ5d (x) |0〉 , (2.10)

where u and d are the up- and down-quark fields respectively. The pion is therefore thelowest energy state that could be created by such an operator. We will revisit this pointwhen we consider the interpolation of meson states in lattice simulations, in section 4.3.

We can also consider combinations of the C, P and T symmetries acting on fields. Forexample, we can construct a CP transformation by combining both C and P transforma-tions. We remark that it is possible to write an additional term in the QCD Lagrangianthat satisfies all the necessary symmetries as discussed in the previous section,

g

32π2θGaµνG̃

aµν , (2.11)

where G̃aµν , the dual field strength tensor, is defined as

G̃aµν = �µνρσGa ρσ. (2.12)

It is notable however that this term violates CP symmetry. However, the parameter θis currently experimentally bound to be θ � 10−9 [37]. This is known as the strong CPproblem [53, 54]. We do not however include this term in our lattice simulations, andthus we treat QCD to be CP -invariant.

One last important remark regarding these symmetries is that any Lorentz-invarianttheory must be invariant under the full CPT symmetry transformation [55]. Hence CPviolation is equivalent to T violation. This is an important result for cosmology; withouta violation of time reversal it would not be possible to generate a matter-antimatterasymmetry in the universe [35].


2.1.1.2 Chiral Symmetry

Another symmetry that is highly relevant to QCD is chiral symmetry. A QCD-like theorywith Nf flavours of massless quarks would have a global SU (Nf )L×SU (Nf )R×U (1)V ×U (1)A global flavour symmetry group. However this is only true at the Lagrangian level;the U (1)A symmetry is anomalous (i.e. the path integral measure is not invariant underthis symmetry) [52].

In QCD we observe light pseudoscalar states, but not light scalar states; from thisobservation we infer that the underlying dynamics of QCD spontaneously break thisgroup down to SU (Nf )V . In any case chiral symmetry is explicitly broken by quarkmasses, which mix the right- and left-handed components of Dirac spinors. However,the spontaneous breaking of chiral symmetry at low energies is much stronger than theexplicit breaking by the quark masses. Consequently, the three lightest quarks u, d ands have masses small enough such that the QCD Lagrangian exhibits an approximateSU (3)L × SU (3)R flavour symmetry. This is the basis of chiral perturbation theory,which we will introduce in section 2.3.

In the case of u and d, we can consider the effect of taking mu = md. In this limitthere exists an SU (2)V flavour symmetry, which we call isospin. For computationalreasons (which we will discuss later in section 4.3.2) it is worthwhile to perform latticeQCD simulations in the limit where isospin is a good symmetry; for many quantities theeffects of isospin breaking produce effects of O (1%) of isospin-conserving effects, hencemay be much smaller than the statistical errors on observables obtained from latticeQCD simulations. In the isospin limit, the up and down quarks form doublets, i.e.

q =

ud

, q = −d

u

. (2.13)We assign as isospin quantum number to the individual quarks corresponding to the I3component of isospin: +1/2 for u and d, −1/2 for u and d.

We remark that in the Standard Model, quarks also interact electromagnetically; thedifferent charges between the u and d quarks also leads to isospin-breaking. However,given that the electromagnetic coupling e is much smaller than the strong coupling gin this low energy regime, it too produces only small corrections to the overall QCDmatrix elements, and thus up to a good precision we may also neglect electromagneticeffects. However it is important to note that modern lattice simulations are now able tocompute QCD matrix elements with sub-percent level errors; for this reason the errorsfrom neglecting isospin-breaking effects are becoming relevant. This is particularly truefor quantities such as the anomalous magnetic moment of the muon [36] or the formfactors involved in K+ → π0`+ν (K`3) decays [56, 57]. Some quantities, such as the


proton-neutron mass splitting [58], are purely isospin-breaking effects, and thus this iscurrently a very active area of research in the lattice community [59–61].

2.1.1.3 Conserved Currents

Chiral symmetry is a continuous symmetry of the QCD action, and hence we can define aconserved current and charge in accordance with Noether’s theorem [62]. Let us considerthe infinitesimal variation of the QCD action under transformations of the form

ψ (x)→ ψ (x) + δψ (x) . (2.14)

For example, the transformation corresponding to a vector symmetry is defined as

δψ (x) = i�aλaψ (x) , δψ (x) = −iψ (x) �aλa, (2.15)

where �a ∈ R is an infinitesimal parameter, and the matrices λa correspond to thegroup associated with the symmetry (λa = 1 for U(1), or λa = T a for SU (Nf )). Thefull vector symmetry holds either for vanishing or degenerate quark masses; hence theSU (2)V isospin symmetry exists for mu = md. It thus follows from Noether’s theoremthat we can define conserved currents associated to these symmetries,

Jaµ (x) = ψ (x) γµλaψ (x) , (2.16)

satisfying ∂µJaµ = 0. This is known as a Ward Identity [63, 64]. Note that the U(1)Vsymmetry holds for arbitrary quark masses; the corresponding conserved charge is baryonnumber.

Let us consider also the axial-vector (or chiral) transformations

δψ (x) = i�aλaγ5ψ (x) , δψ (x) = iψ (x) γ5�aλa. (2.17)

In the limit of massless quarks, Eq. (2.17) would be a good symmetry. It thus followsthat the quantity ∂µAaµ must be proportional to the quark masses. To show this moreexplicitly, we consider the variation of the action, S, and operators, O, under this sym-metry [65]. The starting point is the path integral, where the expectation value of theoperator O is defined as

〈O〉 =ˆD[ψ, ψ̄, A

]OeiS . (2.18)

If we consider the variation of 〈O〉 under the transformation Eq. (2.17), we obtain

− i 〈δSO〉 = 〈δO〉 , (2.19)


where δS is the variation of the action under the axial transformation, and δO is thevariation of O. For the variation of the action we find

δS =

ˆd4x �a

[−∂µAaµ + {Mq, P a}

]. (2.20)

with Mq being the quark mass matrix,

Aaµ = ψ (x) γµγ5λ

aψ (x) (2.21)

being the partially-conserved axial current, and

P a = ψ (x) γ5λaψ (x) (2.22)

being the pseudoscalar density. This leads to an important result known as the partiallyconserved axial current (PCAC) relation, which (assuming degenerate quark masses ofmass m) may be written as

〈∂µAaµ (x)O

〉= 2m 〈P a (x)O〉 . (2.23)

It is thus clear that in the limit of vanishing quark masses, this current is exactly con-served.

2.1.2 Electroweak Theory

While QCD corresponds to the SU(3) gauge symmetry of the Standard Model, theSU(2)×U(1) gauge group corresponds to the electroweak interaction. This gauge groupis notably broken down to U(1) by the Higgs mechanism [66–68].

The Lagrangian for the electroweak interaction can be summarised as

LEW = Lgauge + Lfermion + LHiggs + LY ukawa. (2.24)

In this section I will discuss the key features of this Lagrangian in turn. The first termdescribes the interaction of the gauge bosons themselves, i.e.

Lgauge = −1

4W aµνW

aµν − 14BµνB

µν , (2.25)

withW aµν = ∂µW

aν − ∂νW aµ +

i

4g2�

abcW bµWcν (2.26)

being the field strength tensor corresponding to the SU(2) group, and

Bµν = ∂µBν − ∂νBµ (2.27)


being the field strength tensor corresponding to U(1). W aµ and g2 are the gauge fields andcoupling constant for SU(2) respectively; similarly Bµ and g1 (which will appear later)for U(1). The Pauli matrices (σa) appearing in Eq. (2.26) are related to the generatorsof SU(2) by T a = σa/2. The generator for the U(1) group is Y , the hypercharge. Thesegenerators satisfy

[T a, Y ] = 0. (2.28)

The gauge bosons couple to the fermionic matter of the Standard Model, described by

Lfermion = iQj��DQj + iuj��Duj + idj��Ddj + iLj��DLj + iej��Dej , (2.29)

where the index j runs over the 3 generations of the fermions. The quark contributionshave been separated into 3 distinct terms: the first describes left-handed quark doublets

Qi =

uLdL

, cL

sL

, tL

bL

, (2.30)plus singlets for the right-handed up-type and down-type quarks,

ui = uR, cR, tR, (2.31)

di = dR, sR, bR. (2.32)

The lepton content is described by left-handed lepton doublets,

Li =

νeeL

, νµ

µ

, ντ

τ

, (2.33)and right-handed singlets,

ei = eR, µR, τR. (2.34)

The quantum numbers of these fermions under the SU(2)× U(1) gauge group are sum-marised in Table 2.1. The electroweak covariant derivative that appears in Eq. (2.29) isgiven by

Dµ = ∂µ −i

2g2σ

aW aµ −i

2g1Y Bµ. (2.35)

At low energies the SU(2) × U(1) group is broken down to U(1); it must therefore bepossible to construct the generator of U(1), Q, from the components of the larger group.


We thus defineeQ = e

(T3 +

Y

2

), (2.36)

where e is the electromagnetic coupling. The electromagnetic coupling cannot be imme-diately read off the Lagrangian; however it is possible to see if we perform a change ofbasis such that Bµ

W 3µ

= cos θW − sin θW

sin θW cos θW

Aµ

Zµ

. (2.37)We may thus identify Aµ as the massless photon, with a coupling

e = g2 sin θW = g1 cos θW . (2.38)

θW is known as the Weinberg angle, and the quantity sin2 θW has been determinedexperimentally to be 0.23120(15) [37].

2.1.2.1 Electroweak Symmetry Breaking

So far we have included all the necessary interactions in the electroweak Lagrangian, buthave not accounted for any mass terms. It is not possible to write down gauge-invariantmass terms for the gauge fields or fermion fields. However, this may be remedied byintroducing a new scalar field, H, which may generate the necessary masses via theHiggs mechanism.

The Lagrangian for the Higgs sector is given by

LHiggs =1

2(DµH)

†DµH − V (H) , (2.39)

where the Higgs potential is defined as

V (H) = λ(H†H

)2+ µ2H†H. (2.40)

The scalar field, H, which we identity as the Higgs field, is an SU (2) doublet,

H =

h+h0

, (2.41)which we would like to acquire a VEV

〈H〉 = 1√2

0v

. (2.42)


We remark that SU (2) × U (1) transformations ensure that we can align the VEV inthe direction shown, such that the electromagnetic group U (1) with generator Q [Eq.(2.36)] will remain.

The SU (2) × U (1) group will be broken if we have µ2 < 0 in Eq. (2.40). The minimaof this potential correspond to vacua with

v =

√−µ2λ

. (2.43)

We can subsequently rewrite our Lagrangian by redefining the Higgs field to be

H = h+ v. (2.44)

Any field that couples to the Higgs will thus gain a mass term associated with the HiggsVEV. For the gauge bosons, these originate from the kinetic term (DµH)†DµH; afterspontaneous symmetry breaking the kinetic term may be written as (Dµh)†Dµh, andwe gain the mass terms

LHiggs ⊃g2v2

4W+W− +

g2v2

8

(Zµ

cos θW

)2, (2.45)

where we have defined W± = (W1 ± iW2) /√

2. We can thus see that we have acquiredthe masses MW = g2v2/4 and MZ = MW / cos θW for the W and Z fields respectively,while there is one remaining massless gauge field: the photon, A.

2.1.2.2 CKM Matrix

The final component of the electroweak Lagrangian Eq. (2.24) is the Yukawa sector.This sector is composed of couplings between fermions and the Higgs, such that fermionsmay acquire masses via the Higgs mechanism. The Yukawa Lagrangian is given by

LY ukawa = −ylijLiHej − ydijQiHdj − yuijQi(iσ2)H∗uj + h.c., (2.46)

where the indices i, j correspond to the three generations of fermions.

After electroweak symmetry breaking, the Yukawa couplings become

Mij =v√2yij . (2.47)

For leptons, the weak eigenstates are the same as the mass eigenstates, and so the matrixMLij must be diagonal. However for quarks the weak eigenstate basis is not equivalent tothe mass eigenstate basis; to recover the correct mass terms for the quarks we therefore


must change basis. We may thus write

qiL/R → UijL/Rq

jL/R, (2.48)

M qij = ULmqijU†R, (2.49)

where mqij is the diagonal mass matrix for either up-type or down-type quarks,

mqij =

mu/d 0 0

0 mc/s 0

0 0 mt/b

, (2.50)

and UL, UR are the matrices corresponding to the change of basis.

We must be consistent with this change of basis across the full Lagrangian; the kineticterms in Eq. (2.29) are unaffected by this transformation. Furthermore, the terms cou-pled to the fields Zµ and Aµ are left invariant. The change is relevant only for theinteraction terms between the W± bosons and quarks, i.e.

e√2 sin θW

QiγµW+µ Qi →

e√2 sin θW

QiγµW+µ VijQj , (2.51)

where

V = (UuL)† UdL ≡

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

(2.52)

is the Cabibbo-Kobayashi-Maskawa (CKM) matrix [69, 70]. The physical interpretationof this change of basis can hence be understood as follows. The quarks will propagatethrough space as their mass eigenstates; however the SU (2) quark doublets that experi-ence the weak interaction are superpositions of these mass eigenstates. We therefore findthat the weak interaction is capable of changing the flavour of a quark. If neutrinos weremassless, there would be no such mixing in the lepton sector: any linear combinationof massless eigenstates is still massless, hence the weak and mass bases are equivalent.We now know that this is not true; the discovery of neutrino oscillation [71–73] impliesthat neutrinos do have masses in nature; the mass eigenstates are again not equivalent tothe weak eigenstates. The PMNS matrix [74, 75] describes the mixing within the leptonsector.

We remark that the change of basis involved a unitary transformation, and thus theCKM matrix is also a unitary matrix. A 3 × 3 unitary matrix has a total of 32 = 9independent real parameters. We may reduce the number of parameters by absorbingfive phases into the quark fields (one might naively expect six, although the sixth phase is


an overall phase and does not change the form of the CKM matrix). Thus CKM matrixhas four independent parameters, which we decompose into three real angles (θ12, θ13,θ23) and one phase (δ). The PDG parametrisation of the CKM matrix is [76]

VCKM =

c12c13 s12c13 s13e

iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13

s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13

, (2.53)

where we have used the notation sij ≡ sin θij and cij ≡ cos θij . The complex phasethat appears in this matrix is notable for being the only source of CP-violation in theStandard Model.

2.2 Fermi Effective Theory

The characteristic energy scale of QCD is ΛQCD = O (200 MeV), i.e. the scale at whichthe theory becomes strongly coupled. At scales sufficiently above ΛQCD, QCD may betreated perturbatively. For contrast, the typical scale of weak interactions is O (MW ),which is significantly above ΛQCD. The highest scales in reach of a lattice simulationare typically O (2 GeV), at such scales it is not possible to resolve such heavy particles.We may thus proceed to "integrate out" these heavy degress of freedom: i.e. formallyremove the heavy bosons of the weak interaction and heavy quarks as dynamical degreesof freedom and consider the appropriate low energy effective theory, where light quarkinteractions mediated by the exchange of heavy particles are described by local operators.The operator product expansion provides us with the theoretical framework to performthis task, which I introduce in this section.

2.2.1 Operator Product Expansion

The idea of the operator product expansion is to factorise out the high-energy (short-distance) and low-energy (long-distance) parts of the theory. Let us consider the exampleof the four-quark interaction describing a qs→ qd transition [77, 78]. Example diagramscontributing to this transition are shown in Fig. 2.1. We would like to match the fulltheory of the weak interaction that describes this onto an appropriate effective theory.i.e.

A =

ˆd4x

∑i

Ci (µ) 〈Qui (µ)−Qci (µ)〉 , (2.54)

where the local operators Qqi (q = u, c) involving the active flavours of quarks areweighted by the Wilson coefficients Ci. We have introduced a dependence on the


q d

qW

s

g

q d

qW

s

Figure 2.1: Diagrams giving rise to Q1 and Q2 (current-current) operators.

renormalisation scale µ, which is the characteristic scale separating the short- and long-distance regimes.

The short-distance behaviour is encoded within the Wilson coefficients, Ci (µ), which maybe computed perturbatively as long as we keep the renormalisation scale µ sufficientlyfar above ΛQCD. The long-distance behaviour is given by the matrix elements of localoperators, Qi (µ). We remark that the final amplitude must be independent of µ; thisdependence therefore cancels between the Wilson coefficients and the matrix elements inthe final result. For the example given here, we have two choices for Qi which differ onlyby their colour structure, i.e.

Qq1 = (siqj)V−A(qjdi

)V−A , (2.55)

Qq2 = (siqi)V−A(qjdj

)V−A , (2.56)

where we have used the notation

(qq)V±A = qγµ (1± γ5) q. (2.57)

The indices i, j are colour indices.

2.2.2 Calculation of Wilson Coefficients

The calculation of Wilson coefficients may be performed in perturbation theory at ascale µ � ΛQCD. The conventional choice of renormalisation scheme for the evalua-tion of Wilson coefficients is the MS scheme [77–79]. We encounter a problem howeverif we take µ � MW , as the perturbative calculation introduces terms with large loga-rithms. For example, while αs (µ) may be small, the combination αs (µ) log

(M2W /µ

2)

is not and thus the perturbative expansion breaks down. To avoid this, we will makeuse of renormalisation group improved perturbation theory. The renormalisation groupdescribes the transformation between theories with different choices of renormalisationscale µ. By solving the renormalisation group equations and running from a high scaleµW ' MW to a low scale µ = O (1 GeV), we sum the logarithmic terms to all orders inperturbation theory. For example, the leading logarithmic approximation (LLA) sumsall terms of the form

(αs (µ) log

(M2W /µ

2))n, n ∈ [0,∞); the next-to-leading logarithmic


q d

qs

g

q d

qs

Figure 2.2: Example diagrams contributing to the calculation of Zij .

approximation (NLLA) extends this to terms of the form αs (µ)(αs (µ) log

(M2W /µ

2))n,

etc.

In the remainder of this section I will detail the evolution of the Wilson coefficients fromthe scale µW down to a scale accessible to lattice simulations. A more comprehensivedescription of this procedure may be found, for example, in Refs. [77, 78].

2.2.2.1 Operator Renormalisation and Mixing

We begin the calculation by computing the Wilson coefficients Ci (µW ), which have beenrenormalised at a high scale µW . The first step is to compute the full Standard Modelamplitude to the desired order in perturbation theory, which is achieved by computingdiagrams such as in Fig. 2.1. This must be matched onto the amplitude computedin the effective theory (to the same order in perturbation theory), which contains theoperators Q1 and Q2. This matching reveals an interesting feature of the calculation.When using the renormalised quark states and vertices, the Standard Model amplitudeis finite. However in the equivalent calculation in the effective theory with renormalisedquark states, where theW propagator has been "pinched" to a point [as in Fig. 2.2], it isdivergent. As these divergences do not appear in the full theory, they must be properlyregulated and removed. This necessitates an additional renormalisation condition for theoperators,

〈Qi〉b = Z−2q Zij 〈Qj〉 , (2.58)

where we have included also the quark renormalisation qb = Z1/2q q (the superscript b

indicates the bare, unrenormalised quantity). We remark that Zij is a matrix, andsubsequently the operators Qi are said to mix under renormalisation. In effect thismeans that divergences in the matrix element 〈Q1〉b are regulated by those in the matrixelement 〈Q2〉b and vice versa, such that both renormalised matrix elements are finite.The Wilson coefficients are thus read off by equating the Standard Model amplitude tothe effective theory amplitude, i.e. by setting

A = CiZ2qZ−1ij 〈Qj〉b . (2.59)


It is also equivalent to consider a picture in which the Wilson coefficients are couplingconstants in front of the bare operators within the effective theory. In such a picture itis these "couplings" that must be renormalised to absorb the divergences. The renor-malisation condition in such a picture is thus

Cbi = Zcij (µ)Cj (µ) . (2.60)

These two interpretations are equivalent, and we find that they are related by

Zcij = Z−1ji . (2.61)

2.2.2.2 Renormalisation Group Evolution

At this stage we have the Wilson coefficients renormalised at a scale µW ; we now mustevolve them down to a lower scale µ at which hadronic matrix elements may be computed.The running of the renormalised Wilson coefficients may be obtained by solving therenormalisation group equations

d

d lnµCi (µ) = γji (αs)Cj (µ) , (2.62)

where the anomalous dimension, γ, is defined as

γij = Z−1ik

d

d lnµZkj , (2.63)

using the renormalisation matrix Zij defined in Eq. (2.58). This equation may be solvedup to some desired order in perturbation theory at a scale µW ' MW . We may writethe solution in terms of an evolution matrix,

Ci (µ) = Uij (µ, µW )Cj (µW ) , (2.64)

where U (µ, µW ) encodes the renormalisation group running from a scale µW to a scaleµ. The calculation of U thus requires solving

Uij (µ, µW ) = Tg exp

[ˆ g(µ)g(µW )

dg′γij (g

′)

β (g′)

], (2.65)

order by order in perturbation theory, where the g-ordering operator Tg is defined as

Tg f (g1) , . . . , f (gn) =∑perm

Θ (gi1 − gi2) Θ (gi2 − gi3) . . .Θ(gin−1 − gin

)f (g1) , . . . , f (gn) ,

(2.66)


with Θ (x− y) being the Heaviside step function and the sum runs over all permutationsof gi. The effect is such that in the Eq. (2.65) the coupling constants increase from rightto left.

2.2.2.3 Flavour Boundaries

One more complication that arises in this calculation is the fact that we must integrateout heavy quarks as we evolve down to the desired scale. At a scale µ < mb, the b-quark no longer behaves like an explicit degree of freedom in the theory and thus may beintegrated out. There thus exists a threshold at which we must match between a theorywith 5 active quark flavours to one with 4 active flavours. At the very least this affectsthe running of αs, hence must be taken into account regardless of the process we arecomputing. Depending upon how low we wish to run, we may also choose to integrateout the charm quark. We should be careful however, as the closer µ is to ΛQCD, thepoorer the perturbative description of the physics becomes. In general, the matchingmatrix M between a f and f − 1 flavour theory is computed by imposing the condition

Cfi (µm)Qfi (µm) = C

f−1i Q

f−1i (µm) (2.67)

at the matching scale µm.

2.3 Chiral Perturbation Theory

At low scales, the quarks and gluons of QCD are no longer useful degrees of freedomto consider; rather they form bound states: mesons and baryons. It is natural thereforeto consider whether it is possible to create a low-energy effective theory of QCD, withhadronic degrees of freedom. In this section I introduce Chiral Perturbation Theory(ChPT), which is a low-energy effective description of the interactions of the pseudoscalarmesons in QCD. Useful reviews may be found in Refs. [80–82]. There are multipleapplications of chiral perturbation theory for QCD; one useful feature is that it is possibleto use ChPT results to extrapolate lattice results from unphysical meson masses tophysical ones. Furthermore ChPT itself has some predictive power; the Lagrangian iswritten in terms of low-energy constants (LECs) that give the strength of the variousinteractions that one can write down. The LECs may only be determined by comparingChPT calculations to either experimental or lattice QCD results. However ChPT maystill make useful predictions if LECs can be measured from different processes or ifit makes useful parametrisations for form factors that can be used as inputs for fitsto experimental data. We do not directly use ChPT in this work; however ChPT isrelevant to previous studies of long-distance contributions rare kaon decays, which willbe discussed in section 3.2.


In section 2.1.1.2 we briefly introduced the concept of chiral symmetry and chiral sym-metry breaking in QCD. The key observation is that the small masses of the u, d ands quarks imply that there exists an approximate SU (3)L × SU (3)R symmetry of theQCD Lagrangian, which is broken down to SU (3)V by QCD dynamics. This sponta-neous symmetry breaking must necessarily result in Nambu-Goldstone bosons (NGBs);one for each of the broken symmetry generators of the residual symmetry. For QCD,these NGBs are the octet of the lightest pseudoscalar mesons, which we can write as

φ =

π0√

2+

η√6

π+ K+

π−−π0√

2+

η√6

K0

K− K0 −2η√

6

∼

uu du su

ud dd sd

us ds ss

. (2.68)

This octet transforms as φ → UφU †, where U ∈ SU (3)V . This symmetry is exact fordegenerate quark masses. Furthermore in the limit of vanishing quark masses the mesonstates would be massless; however in nature the quark masses explicitly break axial vectorsymmetries, i.e. they explicitly break SU(3)L × SU(3)R as the left- and right-handedquarks transform differently (electromagnetism also breaks the flavour symmetry). Thepseudoscalar mesons thus acquire masses, and hence they are referred to as pseudo-Nambu-Goldstone bosons (pNGBs). In reality, the pion has a rather small mass of∼ 140 MeV, whereas the kaon has a much larger mass of ∼ 500 MeV. Because wehave ms � md ' mu, we expect that Nf = 3 ChPT is naturally less accurate thanNf = 2 ChPT. Phenomenologically however, Nf = 3 ChPT is far more useful, as it hasapplication to a much larger set of meson decays and interactions.

The pseudoscalar mesons in this octet are much lighter than other hadronic states. Forexample, the ρ meson is the next-heaviest meson, which has a mass of mρ = 770MeV.This suggests therefore that the characteristic scale of chiral symmetry breaking is onthe order of 1GeV; below this scale we may hence consider an effective theory in whichthis octet of pseudoscalar mesons are the only degrees of freedom. The Lagrangian of theeffective theory is constructed using symmetry arguments; this "bottom-up" approachcontrasts with the "top-down" approach of the operator product expansion presented insection 2.2. The effective Lagrangian must be invariant under chiral symmetry; whichcan be written in terms of the unitary field

Σ = exp

(2iφ

f

), (2.69)

where φ is the field defined in Eq. (2.68) and f is a dimensionful parameter such thatφ has the canonical mass dimension of a scalar field, which can be identified as the piondecay constant. In order to write down a Lagrangian in terms of this field, note that wecannot have terms of the form ΣΣ† = 1. In general, to form a term that is both Lorentzand SU (3) invariant, we must take the trace of products of derivatives of Σ and Σ†,


and each term must have an equal number of derivatives of each. Because each term inthe Lagrangian involves derivatives of the field, the amplitudes of the interactions theydescribe must vanish for vanishing external momenta. Hence the strength of the pseu-doscalar meson interactions are proportional to the magnitude of the external momenta.We must identity the expansion parameter for the theory as (p/f), and only even powersappear in the expansion.

Before we write down the Lagrangian for ChPT, we must also understand how quarkmasses are introduced into the theory, given that they are forbidden by chiral symmetry.The general form for the quark mass terms is qMq, where M is a matrix with the quarkmasses md, mu, ms respectively on the diagonal. Instead of writing a mass term directlyinto the ChPT Lagrangian, we may introduce a new ‘spurion’ field χ, which transformsas

χ→ ULχU †R, χ† → URχ†U†L (2.70)

under a chiral transformation. We thus recover the mass terms by setting χ = χ† =2B0M . Here B0 is some a priori unknown LEC. The Lagrangian of ChPT at leadingorder, O

(p2), is thus

L(2)eff =1

4f2Tr

(∂µΣ

†∂µΣ)

+1

4f2Tr

(χ†Σ + Σ†χ

). (2.71)

We remark that this Lagrangian may be used to compute only tree level scatteringamplitudes of pseudoscalar mesons. Higher order (loop) corrections enter at O

(p4)in

the chiral expansion; for a consistent calculation at this order we must also considerthe additional terms involving four derivatives that may enter the chiral Lagrangian,which effectively lead to new tree level contributions. We remark that the theory isrenormalisable order by order; the additional tree level contributions introduced atO

(p4)

act as counterterms to remove divergences from the loop corrections constructed fromL(2)eff .

In addition to describing hadronic interactions, we must encode weak and electromagneticinteractions into the chiral Lagrangian in order to describe (semi-)leptonic or non-leptonicweak decays involving the transition s → d. Such a Lagrangian is therefore applicableto studies of the rare kaon decays K → π`+`−, which will be discussed in section 3.2.1.To lowest order, the weak contributions can be written as [20]

L(2)∆S=1 =GF√

2VudV

∗usG8 (LµL

µ)23 , (2.72)

where

Lµ = if2Σ∂µΣ

†, (2.73)


and the subscript 23 indicates the relevant entry of the SU(3) matrix (i.e. the entrycorresponding to an s→ d transition). This term thus describes the left-handed currentof the weak interaction. Electromagnetism may be included as in the Standard Modelvia a U (1) gauge interaction. We must therefore replace derivatives with their covariantcounterparts, which generates the additional terms

L(2)EM = −eAµTr(Q̂V µ

)+ e2AµA

µ 1

2f2(

1− |Σ11|2), (2.74)

where Aµ is the gauge field of the photon and Q̂ = diag (1, 0, 0) is the generator of theU(1

eprints.soton.ac.uk · 2019. 3. 13. · universityofsouthampton abstract...

Documents